The University of British Columbia November 9th, 2017 Midterm for MATH 104, Section 101 : Solutions

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1 The University of British Columbia November 9th, 2017 Mierm for MATH 104, Section 101 : Solutions Closed book examination Time: 50 minutes Last Name First Signature Student Number Section Number: Special Instructions: No memory aids are allowed. No calculators. No communication or other electronic devices. Show all your work; little or no credit will be given for a numerical answer without the correct accompanying work. If you need more space than the space provided, use the back of the previous page. Where boxes are provided for answers, put your final answers in them. Mierms written in pencil will not be considered for regrading. Rules governing examinations Each candidate must be prepared to produce, upon request, a UBCcard for identification. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness shall not be received. Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator. Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator Total 50 Page 1 of 9

2 November 9th, 2017 Math 104 Name: Page 2 of 9 1. Short Answer Questions [20 points]. Put your final answer in the box provided, but NO CREDIT will be given for the answer without the correct accompanying work. (a) [3 points] Find when cos(xy) = x + y. Solution : We have, differentiating implicitly, ( sin(xy) y + x ) and so = 1 + = y sin(xy) 1 x sin(xy) + 1. (b) [3 points] Find the equation of the normal line to the curve at the point (x, y) = ( 2, 1). xy + x 2 1 = y Solution : We have y + x + 2x = and so, for x = 2 and y = 1, we have that = 1. It follows that the normal line at this point has slope 1/ = 1 and hence equation y 1 = 1(x + 2) = x + 2. (c) [3 points] Let f(x) = x 1 x for x > 0. Does f(x) have a local maximum? If so, at which value of x? Solution : Taking logarithms and differentiating, we have log f(x) = log x x and so f 1 (x) f(x) = x log(x) 1 x = 1 log x. x 2 x 2 Since f(x) = x x > 0 for all x > 0, it follows that f (x) is positive provided log x < 1, negative of log x > 1 and equal to zero if log x = 1, i.e. if x = e. The first derivative test thus implies that f has a local maximum at x = e. (d) [3 points] How much initial investment is required to generate $1000 in interest over a period of 2 years if the interest rate is 5% per annum, compounded annually? Do not try to simplify your answer. Solution : Given an initial investment of P 0 dollars, the total amount in ten investment after t years will be P (t) = P 0 ( ) t. We thus want to solve the equation We find that dollars. P (2) = P , i.e. P 0 (1.05) 2 = P P 0 =

3 November 9th, 2017 Math 104 Name: Page 3 of 9 (e) [4 points] If the base b of a triangle is increasing at a rate of 3 centimetres per second, while its height h is decreasing at a rate of 3 centimetres per second, which of the following must be true about the area A of the triangle? (A) A is always constant. (B) A is always decreasing. (C) A is always increasing. (D) A is decreasing only when b < h. (E) A is decreasing only when b > h. Solution : We have A = bh/2, so that If we have db = 3 and dh da = 1 2 = 3, it follows that ( b dh ) + hdb. da = 3 (h b) 2 and so A is decreasing precisely when da < 0 and hence only when b > h. (f) [4 points] Let f(x) be continuous and differentiable everywhere on the closed interval [0, 10]. Suppose that f(0) = f(10) = 0 and f(5) = 4. Which one of the following is not necessarily true: (A) There is some c (0, 10) such that f(c) is a global maximum. (B) There is some c (0, 10) such that x = c is a critical point. (C) There is some c (0, 5) such that f(c) is a local minimum. (D) There is some c (0, 5) such that f(c) = 2. (E) There is some c (0, 5) such that f (c) = 4/5. Solution : (A) follows from continuity, (B) from Rolle s Theorem, (D) from the Intermediate Value Theorem and (E) from the Mean Value Theorem. (C) can fail to happen when, for example, f is strictly increasing between x = 0 and x = 5.

4 November 9th, 2017 Math 104 Name: Page 4 of 9 2 [10 points]. The demand curve of a certain product is given by p 2 + 4q + pq = 10, where p is the price in dollars and q is in thousand units. The price elasticity of demand is ε(p) = p dq. q dp (a) [5 points] Compute the price elasticity of demand ε(p) when the price is p = $2. Solution : Differentiating implicitly, 2p + 4 dq dp + q + pdq dp = 0. Substituting p = 2, we find that q = 1 and so and so dq dp = 5/6. It follows that dq dp dq dp = 0 ε = p q dq dp = 5/3. (b) [2 points] If the price is increased from $2 by 3%, what is the percentage change in demand? Solution : Since 5/3 3 = 5, we have a 5% decrease in demand. (c) [3 points] Does revenue increase or decrease when the price is increased from $2 by 3%? Justify your answer. Solution : Since 5/3 > 1, revenue will decrease when price is increased from $2.

5 November 9th, 2017 Math 104 Name: Page 5 of 9 3 [10 points]. A bug starts at the origin (x, y) = (0, 0) and walks towards the point (x, y) = (10, 100) along the parabola y = x 2 in such a way that its distance to the origin increases at a rate of 1 centimetre per minute. Determine the bug s horizontal speed and vertical speed when it is at the point (x, y) = (2, 4). Solution : If D is the distance from a point (x, y) to the origin, then and so D 2 = x 2 + y 2, 2D dd = 2x + 2y. When (x, y) = (2, 4), we find that D = 20 = 2 5 and so our equation becomes, when we include the information that dd = 1, Since y = x 2, we have that = 2x 5 = + 2. and hence, when x = 2, = 4. Substituting this, 5 = + 8 = 9 whereby 5 = cm per minute, and 9 = 4 = 4 5 cm per minute. 9 4 [10 points]. You are standing on the bank of a calm river that is 100 metres wide and runs East-West, and see your friend waving to you from the opposite bank of the river, 300 metres east of you (so that their straight-line distance away from you is metres). You can swim at 3 km/hour (i.e m/hr) and walk at 5 km/hour (i.e m/hr) and you want to get to your friend as quickly as possible. To what point on the opposite shore should you swim before walking the rest of the way? Solution : We will swim to a point x metres downstream of our current location on the other side of the river and then walk the remaining 300 x metres (see Figure 1 on the next page). By the Pythagorean theorem, the distance swum is x metres. The time taken to swim and walk is therefore x T (x) = x Here, we have 0 x 300. We have T (x) = 1 x 3000 x and so T has no singular points and a single critical point corresponding to where T (x) = 0, i.e. when x x =

6 Figure 1: swimproblem Solving this, we find that We thus have 5x = 3 x and so 25x 2 = 9x x 2 = and so x = 75 (we can omit the case x = 75). To see that swimming to this point minimizes time, we can compute T (x) = 1 x x x x = 1 x x x (x ) = 10 3/2 3(x ). 3/2 Since this is always positive, our critical point must correspond to an absolute minimum, by the second derivative test.